Hell yeah, fat guy touchdowns are always the best.
https://www.youtube.com/watch?v=QXxsz4Bxn1IThat’s a good one and it always worked for Buck Rogers when he was in a fight.
In the last Superbowl the Patriots ran 3 2-point plays (the third one was the last play of the game - they decided to use the last 2-point play from their call sheet for that play). One was a direct snap to the RB, one was a quick pass to a WR, the third was a sweep.
I’d go with the direct snap to a RB.
If they’d had Gronk they’da thrown to Gronk. Or not been in that hole to start with.
5 OL, QB & RB in pistol formation. Out wide to one side, 3 blockers (eligible as necessary) on the line, with my best big receiver (a Gronk or Megatron would be great) behind them. Based on how the D lines up…
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hand off up the middle for a leap over the pile (this being the safer option)
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quick out to the receiver standing still and then fight for that yard.
The more defenders you can get away from the center of the formation, the better. But if they only send 4 to the right, I like my big WRs chances of pushing forward for just a yard against safeties and corners.
Edit - the left tackle here will need to be eligible, of course. Line up as necessary. Feel free to move 1 TE wide left if you prefer.
Lots of fun, but it deprived a much more deserving Walter Payton of a chance to score a Super Bowl TD.
You don’t see it much anymore, but I’d hand off to a guy who can jump over the line and land in the end zone.
I’d have no idea for a play. If my Browns were ever in this position, I’d be passed out from doing shots for the past hour.
Just don’t hand off to Earnest Byner. 
A guy walks into a bar in Cleveland with a dog. The dog promptly jumps up on the bar top, and the bartender comes over.
“Hey, you can’t have your dog in here!”
Just then, on the TV behind the bar, the Browns kick a field goal. The dog barks at the TV and does three back flips on the bar. The bartender says “Cool, three back flips for 3 points. I like that. You and your dog can stay. What does he do when the Browns get a touchdown?”
The man answers “I don’t know, he’s only 2 years old.”
QB rollout to his good side. An RB there as an option and 2 WR + 1 TE on that side.
I’m remembering what Bob Devaney did in a situation like this when his Nebraska Cornhuskers played Oklahoma for the #1 spot going into Bowl season. (There was more time on the clock, but it was obviously a game situation.) He called his QB Jerry Tagge over and asked what play had been working best. Tagge responded that I-Back off tackle hadn’t failed all day, so that’s what Devaney called. It worked, and Nebraska went on to beat Alabama for the National Championship.
I don’t know if it would ever be that informal today - any major program has someone upstairs keeping track of every play. The principle would be the same, though: is there a play they haven’t been able to stop? Go with that.
Unless it’s a draw. I saw Ty Willingham call for that on 4th and 2 one time against Florida State. It was very obviously the wrong play to run in that situation.
Yeah, a draw play takes time, and also lets defenders into your backfield. You just can’t let that happen in a do-or-die on the last play of the game from the 1. You don’t have the luxury of time or nonchalance like that; this takes urgency.
It doesn’t matter. Whatever they call it will result in a loss of 6 yards.
Tight end reverse, flip back to QB:
That doesn’t count. It was from the 2 yard line. 
I still recall a Browns/Ravens game (from last season?) where the score was tied, there were seconds left, and the Browns opt for a field goal around the 30 yard line. Nothing to lose, either it misses and they go into overtime, or they make it and win.
Being the Browns, the kick is blocked, recovered by the Ravens, and run down the field for a touchdown.
Never underestimate the Browns. They will always find a way.
ETA: I looked it up, it was in December 2015.
First you get the Fat Guy Catch, then you get the Fat Guy Run, then you get the Fat Guy Dance.
Followed by the Fat Guy sits on the bench sucking on an oxygen mask for 5 mins. 
Yeah, Saint Cad, see? The Browns don’t always get a six-yard loss. There are other options.
Just run it in and let’s get the hell out of here.
A fullback trap would be my gut-level idea, yeah, especially if I have a guy who’s darn near impossible to bring down immediately. Someone like Christian Okoye or Bronko Nagurski.
Can I bring The Fridge out of retirement?
Because he’s going over!